Covering a compact space by fixed-radius or growing random balls

TL;DR

This paper extends random coverage models to general compact metric spaces, providing bounds on cover times for fixed-radius and growing balls, with implications for Markov chains and stochastic processes.

Contribution

It introduces new bounds for coverage by random sets on compact spaces and analyzes the growth model, connecting geometric coverage with Markov chain hitting times.

Findings

Weak concentration bounds for cover times established

Coverage by i.i.d. sets with arbitrary distribution analyzed

Growing ball model offers tractable analysis and open problems

Abstract

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model.